Experimental design Based on Chapter 2 of D. Heath (1995). An Introduction to Experimental Design...

Experimental design

Based on Chapter 2 of D. Heath (1995). An Introduction to Experimental Design and

Statistics for Biology. CRC Press.

Four critical features of experimental designHurlbert 1984

• Controls

• Randomization

• Replication

• Interspersion

Possible explanations?Research hypothesis

(or hypotheses)

The design of a experiment

• Factor: humidity

• Variable: direction

Removing other possible effects

• Dealing with bias

Other design issues

• Number of woodlice

• Which woodlice

• They must be representative of the population of reference

Confounding factors

Independent observations

Analysis

• Null hypothesis:

• Alternative hypothesis:

Probability of damp turn = 0.5

Probability of damp turn = 0.5

Expected frequencies for four trails

dry

dry damp

dry damp

dry damp

dry damp

dry damp

dry damp

dry damp

damp

dry damp

dry damp

dry damp

dry damp

dry damp

dry damp

dry damp

Example

• Damp*Damp*Damp*Damp• If order does not matter there is only one way to

obtain four damp turns and the combined probability (under the assumption of independence) is 0.5*0.5*0.5*0.5= 0.0625

• Calculate the probability of the other possible outcomes under the null hypothesis

Exercise • There are four ways to obtain three damp turns:

Damp*Damp*Damp*DryDamp*Damp*Dry*DampDamp*Dry*Damp*DampDry*Damp*Damp*Damp

• and the combined probability (under the assumption of independence) is 0.5*0.5*0.5*0.5= 0.0625 four times = 0.25

• Calculate the probability of the other possible outcomes under the null hypothesis

Binomial distribution (4 trials)Under the null hypothesis

0.000.050.100.150.200.250.300.350.40

0 1 2 3 4

Number of damp turns

Expected frequency

Distribution under the null hypothesis(17 trials)

0.00

0.05

0.10

0.15

0.20

0:17 2:15 4:13 6:11 8:09 10:07 12:05 14:03 16:01

Number of damp:dry turns

Expected frequency

What do you conclude if we observed 14 damp turns out of 17 ?

Binomial distribution

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 2 4 6 8 10 12 14 16

Number of damp turns

Rejection region Rejection region

0.0000+0.0001+0.0010+0.0052+0.0182=

2.45%

0.0182+0.0052+0.0010+0.0001+0.0000=

2.45%

likelyunlikely unlikely

Why we start with the null hypothesis?

The main points

• Use a mathematical model to produce a sampling distribution of all possible values of the test statistic assuming that the null hypothesis is true

• Find the probability associated with a a particular value occurring in a particular experiment

• Use the probability to make a decision about whether a particular result is likely or unlikely

The Binomial DistributionOverview

• However, if order is not important, then

where is the number of ways to obtain X successes

in n trials, and n! = n (n – 1) (n – 2) … 2 1

n!

X!(n – X)! ppXX q qn – n –

XX

PP((XX) =) =

n!

X!(n – X)!